In 1982, Richard Feynman proposed that a controllable quantum system could be used to simulate other quantum systems more efficiently than conventional computers. See Feynman, 1982, International Journal of Theoretical Physics 21, pp. 467-488. This controllable quantum system is now commonly referred to as a quantum computer, and effort has been put into developing a general purpose quantum computer that can be used to simulate quantum systems or run specialized quantum algorithms. In particular, solving a model for the behavior of a quantum system commonly involves solving a differential equation related to the Hamiltonian of the quantum system. David Deutsch observed that a quantum system could be used to yield a time savings, later shown to be an exponential time savings, in certain computations. If one had a problem, modeled in the form of an equation that represented the Hamiltonian of the quantum system, the behavior of the system could provide information regarding the solutions to the equation. See Deutsch, 1985, Proceedings of the Royal Society of London A 400, pp. 97-117.
One limitation in the quantum computing art is the identification of systems that can support quantum computation. As detailed in the following sections, a qubit, which is analogous to a “bit” of a classical digital computer, serves as the basis for storing quantum information. However, qubits must be able to retain coherent quantum behavior long enough to perform quantum computations. The loss of coherent quantum behavior is referred to as decoherence. Further, techniques for reading the state of qubits are needed in order to determine the result of a quantum computation. Ideally, such readout mechanisms do not introduce decoherence to the quantum computing system prior to a readout operation.
The computing power of a quantum computer increases as its basic building blocks, qubits, are coupled together in such a way that the quantum state of one qubit affects the quantum state of each of the qubits to which it is coupled. This form of coupling includes the effect referred to as entanglement. Another limitation in the quantum computing art is the identification of methods that can be used to controllably entangle the states of qubits without introducing a significant source of decoherence.
Approaches to Quantum Computing
There are several general approaches to the design and operation of a quantum computer. One approach referred to as “circuit model quantum computing” is based on a model in which logical gates are applied to qubits, much like bits, and can be programmed to perform calculations using quantum logic. This model of quantum computing requires qubits with long coherence times. Efforts have made to develop robust qubits that can perform quantum logic functions. For example, see Shor, 2001, arXiv.org: quant-ph/0005003. However, reducing qubit decoherence in quantum systems to the point that many calculations are performed before quantum information stored in the quantum system is destroyed has not been satisfactorily achieved in the art.
Another approach to quantum computing known as “thermally-assisted adiabatic quantum computing,” involves finding the lowest energy configuration of an array of qubits. This approach does not make critical use of quantum gates and circuits. Instead, it uses classical effects, and quantum effects in some cases, to manipulate the states of a system of interacting qubits starting from a known initial Hamiltonian so that the final state represents the Hamiltonian of the physical system in question. In this process, quantum coherence is not a strict requirement for the qubits. An example of this type of approach is adiabatic quantum computing. See, for example, Farhi et al., 2001, Science 292, pp. 472-476.
Qubits
A quantum bit, or qubit, is the quantum mechanical analog of the conventional digital bit. Instead of only encoding one of two discrete states, like “0” and “1” in a bit, a qubit can also be placed in a superposition of 0 and 1. That is, the qubit can exist in both the “0” and “1” state at the same time, and can thus perform a quantum computation on both states simultaneously. Thus, a qubit holding a pure discrete state (0 or 1) is said to be in a classical state, whereas a qubit holding a superposition of states is said to be in a quantum state. In general, N qubits can be in a superposition of 2N states. Quantum algorithms make use of the superposition property to speed up certain computations.
In standard notation, the basis states of a qubit are referred to as the |0> and |1> states. During quantum computation, the state of a qubit, in general, is a superposition of basis states so that the qubit has a nonzero probability of occupying the |0> basis state and a simultaneous nonzero probability of occupying the |1> basis state. Mathematically, a superposition of basis states means that the overall state of the qubit, denoted |Ψ>, has the form |Ψ>=a|0>+b|1>, where a and b are coefficients corresponding to the probabilities |a|2 and |b|2 of obtaining a |0> or |1> upon measurement, respectively. Coefficients a and b each have real and imaginary components. The quantum nature of a qubit is largely derived from its ability to form a coherent superposition of basis states. A qubit is in a coherent superposition as long as the amplitudes and phases of coefficients a and b are not affected by the outside environment. A qubit will retain this ability to exist as a coherent superposition of basis states when the qubit is sufficiently isolated from sources of decoherence.
To complete a computation using a qubit, the state of the qubit is measured (e.g., read out). Typically, when a measurement of the qubit is done, the quantum nature of the qubit is temporarily lost and the superposition of basis states collapses to either the |0> basis state or the |1> basis state, thus regaining its similarity to a conventional bit. The actual state of the qubit after it has collapsed depends on the probabilities |a|2 and |b|2 immediately prior to the readout operation. For more information on qubits, generally, see Nielsen and Chuang, 2000, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, pp. 344-345.
Superconducting Qubits
There are many different technologies that can be used to build quantum computers. One implementation uses superconducting materials. Superconducting qubits have the advantage of scalability. The possibility of realizing large scale quantum computers using superconducting qubits is promising since the technologies and processes involved in fabricating superconducting qubits are similar to those used for conventional silicon-based computers, for which there already exists infrastructure and technological know-how. Toward the realization of such a computer, Shnirman et al., 1997, Physical Review Letters 79, 2371-2374, proposed a superconducting quantum computer using Josephson junctions to produce the required quantum effects.
Superconducting qubits can be separated into several categories depending on the physical property used to encode information. A general division of qubits separates them into charge and phase devices, as discussed in Makhlin et al., 2001, Reviews of Modern Physics 73, pp. 357-400.
A superconducting qubit is typically characterized by two different types of energy, charging energy Ec, and Josephson energy EJ. The magnitude of each of these energy types in a given superconducting qubit depends on the physical parameters of the qubit. For example, the charging energy of a superconducting qubit is a function of the charging energies of the components (e.g., qubit junctions) of the qubit. The charging energy of a qubit junction, in turn, is defined as e2/(2C), where C is the capacitance of the junction. The Josephson energy of a superconducting qubit is a function of the Josephson energies of the components (e.g., qubit junctions) in the qubit. The Josephson energy of a qubit junction (e.g., Josephson junction), in turn, is related to the critical current of the qubit junction. Specifically, the Josephson energy of a qubit junction is proportional to the critical current IC of the junction and satisfies the relationship EJ=(ℏ/2e)Ic, where ℏ is Planck's constant divided by 2π. The ratio of the overall Josephson energy and the overall charging energy of a superconducting qubit can be used to classify superconducting qubits. For example, in one classification scheme, when the overall charging energy of a given superconducting qubit is much greater than the overall Josephson energy of the qubit, the qubit is deemed to be a charge qubit. And, when the overall Josephson energy of a given superconducting qubit is much greater than the overall charging energy of the qubit, the qubit is deemed to be a phase qubit. As used herein, the term “much greater” in the context of evaluating two energy terms means that one energy term may be anywhere from two times greater to more than twenty times greater than the second energy term.
In quantum systems based on qubits, phase and charge are conjugate variables. That is, a higher accuracy of determination of the phase leads to a greater uncertainty in the charge and vice versa. Charge qubits are said to operate in the charge basis (or regime), where the value of the charge is more localized, while phase qubits operate in the phase basis, where the value of the phase is more localized.
Charge qubits store and manipulate information in the charge states of the device, where elementary charges consist of pairs of electrons called Cooper pairs. A Cooper pair has a charge of 2e, where e is the elementary charge, and consists of two electrons bound together by a phonon interaction. See, for example, Nielsen and Chuang, 2000, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, pp. 344-345.
Phase qubits, on the other hand, store information in the phase or flux states of the qubit. Phase qubits include a superconducting loop interrupted by a Josephson junction. Phase qubits can be further distinguished as either flux qubits or “phase-only” qubits. Flux qubits are characterized by relatively large superconducting loops that can trap large fluxes on the order of the unit flux Φ0=hc/2e. See Bocko et al., 1997, IEEE Trans. Appl. Superconduct. 7 3638. “Phase-only” qubits, on the other hand, are characterized by a small inductance and are magnetically inactive. A “phase-only” qubit stores information in the form of a phase drop across a Josephson junction interrupting the superconducting loop. See, for example, loffe et al., 1999, Nature 398, 679.
Another type of qubit is the hybrid qubit. Hybrid qubits use both the charge and phase degrees of freedom to control information. Some examples of hybrid qubits are described in U.S. Pat. No. 6,838,694; and United States Patent Publication No. 2005-0082519, which are hereby incorporated by reference in their entireties.
Superconducting Flux Qubits
One proposal to build a quantum computer from superconducting qubits is Bocko et al., 1997, IEEE Transactions on Applied Superconductivity 7, p. 3638. See also, Makhlin et al., 2001, Review of Modern Physics 73, p. 357-400. Since then, many designs have been introduced. One such design is the persistent current qubit. The persistent current qubit is a form of flux qubit, meaning that it is a phase qubit that can store fluxes on the order of the unit flux Φ0=hc/2e. See Mooij et al., 1999, Science 285, 1036; and Orlando et al., 1999, Physics Review B 60, 15398. As illustrated in FIG. 6, the persistent current qubit comprises a loop of thick superconducting material interrupted by three small-capacitance Josephson junctions (denoted as “X” in FIG. 6) in series. The superconducting loop can enclose an applied magnetic flux fΦO, wherein ΦO is the superconducting-flux quantum h/2e, where h is Plank's constant. The value of the coefficient f can be controlled by an external magnetic bias and is usually kept at a value slightly smaller than 0.5. The critical current value of one Josephson junction, denoted aEJ in FIG. 6, is engineered to be less than that of the critical current value EJ of the other two Josephson junctions, which often have the same or very similar critical currents (which values are each denoted EJ in FIG. 6). Typically, a is in the range 0<a<1. The persistent current qubit can be built such that the loop of superconducting material encloses a small area, (e.g., less than ten microns squared).
The persistent current qubit is well known and has demonstrated long coherence times. See, for example, Orlando et al.; and Il'ichev et al., 2003, Physics Review Letters 91, 097906. Some other types of flux qubits comprise superconducting loops having more or fewer than three Josephson junctions. See, e.g., Blatter et al., 2001, Physics Review B 63, 174511; and Friedman et al., 2000, Nature 406, 43.
The sign of the coupling interaction in the system Hamiltonian that describes the coupling of two superconducting flux qubits can be used as a basis for classifying qubit coupling types. According to such a classification scheme, there are two coupling types, ferromagnetic and anti-ferromagnetic.
Flux qubits typically interact via their respective magnetic fluxes. That is, a change in flux in a first superconducting flux qubit will cause a change in flux in a second superconducting flux qubit that is coupled to the first superconducting flux qubit. In ferromagnetic coupling, it is energetically favorable for a change in flux of the first superconducting flux qubit to produce a similar change in the flux of a second superconducting flux qubit to which the first superconducting flux qubit is coupled. For example, an increase in flux in the first qubit will cause an increase in flux in the second qubit when the two qubits are ferromagnetically coupled. Since circulating loop currents generate flux within the superconducting loop of a flux qubit, ferromagnetic coupling can also mean that circulating current in one qubit will generate current flowing in the same direction in another qubit.
In the anti-ferromagnetic case, it is energetically favorable for a change in flux of a first superconducting flux qubit to produce a similar but opposite change in flux in a second superconducting flux qubit to which the first superconducting flux qubit is coupled. For example, a flux increase in one qubit leads to a flux decrease in the anti-ferromagnetically coupled device. Likewise, a circulating current in one direction in a first flux qubit causes a current flow in the opposite direction in the flux qubit that is anti-ferromagnetically coupled to the first qubit because it is more energetically favorable. By energetically favorable, it is meant that the system comprising the coupled qubits prefers to be in a specific coupling configuration (because the overall energy of the coupled system is lower in the specific configuration than in other configurations).
In the Hamiltonian of two flux devices coupled together, σz{circle around (X)}σz represents the “sigma z” coupling between two devices with a variable J as a pre-factor that indicates the strength of the coupling. When J>0, the coupling is anti-ferromagnetic, with a higher J meaning a stronger anti-ferromagnetic coupling. When J<0, the coupling is ferromagnetic, with a lower J meaning a stronger ferromagnetic coupling. When J=0, there is no coupling. Thus, switching the sign of J switches the type of coupling from ferromagnetic to anti-ferromagnetic or vice versa.
Measurement Techniques for Qubits
Generally, qubit measurement is conducted based on the assumption that the qubit can be in a quantum state. However, qubits can be restricted to hold only classical states and then measured when in this restricted state. Regardless of whether measurement relies on the assumption that the qubits to be measured are in a quantum state or on the assumption that they have been restricted to a classical state, methods and structures in the art that can measure a large number of qubits in the same circuit are lacking. Usually, a readout mechanism for one qubit requires a certain amount of circuit board space, as well as at least one control wire to operate the mechanism. Traditionally, for every additional qubit in a circuit, an additional readout mechanism for that qubit is used, as well as at least one additional control wire. This creates a problem in circuit design when a large number of qubits are present, since space constraints make placement of qubits in a circuit very complex. Also, the presence of additional control wires creates a problem in finding an efficient routing of all the wires in the circuit. In an array with a large number of qubits, reading out the qubits in the interior of the array can be challenging due to restrictions in area and wiring paths into the interior of the array.
Il'ichev et al., referenced above, proposed a method to read out the state of a flux qubit by weakly coupling the flux qubit to a tank circuit. When the qubit is ready for measurement, the qubit is brought into resonance with the tank circuit so that the state of the qubit and the state of the tank circuit couple. The tank is then decoupled from the qubit. This method, although it reduces dissipation of the qubit by the tank circuit when not reading out, is not scalable to higher numbers of qubits in a quantum circuit, because having a single tank circuit for each qubit is not feasible.
One way of measuring a flux qubit is through the use of a superconducting quantum interference device, or SQUID, inductively coupled to the flux qubit. A SQUID comprises a superconducting loop interrupted by at least one Josephson junction. The current flowing in the loop of the SQUID can be biased in several different ways. Two examples of SQUIDs that differ in the way they are biased are dc-SQUIDs and rf-SQUIDs. Since flux devices interact via their magnetic fluxes, a SQUID-type device can be used to couple flux qubits together, like the scheme suggested by Majer et al., 2003, arXiv.org:cond-mat/0308192. When used to measure the state of a flux qubit, the SQUID's supercurrent is read out because this supercurrent is dependent on the state of the qubit. As such, a measurement of the SQUID's current can determine the state of the qubit to which the SQUID is coupled. However, SQUIDs have the drawback that they take up a considerable amount of surface area on a circuit board or chip. For higher numbers of qubits, having a SQUID for each qubit becomes cumbersome and space consuming.
Paternostro et al., 2005, Physical Review A 71, 042311, (hereinafter “Paternostro”) disclose a method of transferring a quantum state of a qubit through a multi-qubit coupling via a bus system. Paternostro combines quantum optics and SQUIDs in order to create a network of spin chains on which quantum operations can be performed. However, including a bus to couple all the qubits together can introduce increased noise interference into the system.